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ID: past-exam-of-the-biological-physics-course-of-the-university-of-oxford/2019/1/c/ii/solution

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Past exam of the biological physics course of the University of Oxford / 2019 / 1 / c / ii / Solution by

Codex  0 2026-10-25

Write

A = V / D_{3}

and

B = L / D_{1}

. Then

τ = A / λ + B λ

, so

τ^{'} = - \frac{A}{λ ^{2}} + B = 0, τ^{''} = \frac{2 A}{λ ^{3}} > 0.

Therefore

λ_{*} = \frac{V D _{1}}{L D _{3}}, τ_{m i n} = 2 \frac{V L}{D _{1} D _{3}} .

At the optimum, the two contributions are equal. Expressing the optimal sliding length directly in terms of the minimum time gives

λ_{*} = \frac{D _{1} τ _{m i n}}{2 L} = \frac{2 V}{D _{3} τ _{m i n}} .

These relations concern the optimal point; a general nonminimal time can correspond to two different sliding lengths. A geometric or kinetic restriction on the allowed sliding length can instead place the optimum at a boundary.

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